Annular seals for non-contact sealing of fluids in turbomachinery

ABSTRACT

According to an aspect of the disclosure, an annular seal generally includes a hollow member having two ends and an inner surface. The hollow member has a number of depressions formed in and extending around its inner surface. Each of the depressions having a depth that is a function of its distance from one of the two ends.

RELATED APPLICATIONS

This patent application is a continuation of co-pending U.S. patentapplication Ser. No. 13/315,973, which was filed on Dec. 9, 2011, whichis a continuation of U.S. patent application Ser. No. 11/744,332, whichwas filed on May 4, 2007, which claims priority to U.S. ProvisionalPatent Application Ser. No. 60/746,582, which was filed on May 5, 2006,the disclosures of which are incorporated herein by reference to theextent consistent with the present application.

TECHNICAL FIELD OF THE DISCLOSURE

This disclosure relates generally to seals, and more particularly, to anannular seal for non-contact sealing of fluids in turbomachinery.

BACKGROUND OF THE DISCLOSURE

Annular seals have been implemented on turbomachinery to prevent leakageof compressible or incompressible fluids such as gases or liquids,respectively. These annular seals generally incorporate a hollow memberhaving a number of depressions formed in its inner surface that isdimensioned to extend around a rotor portion having a relatively smoothsurface. In operation, the physical relationship of the rotor surface tothe stator surface causes a dynamic resistance to the movement of thefluid through the seal such that a sealing action occurs.

Common types of annular seals include hole pattern (HP) seals andhoneycomb (HC) seals. Hole pattern seals generally include depressionsin the form of a number of generally round-shaped holes that are formedin the inner surface of its stator portion. Honeycomb seals generallyinclude depressions in the form of generally hexagonal-shaped holes thatare formed into a lattice-shaped structure. The inner surface of thehollow member is configured in a spaced apart relation relative to theouter surface of the rotor portion so that no contact is made betweenthe hollow member and rotor portion. In this manner, relatively littlewear may be caused to the inner surface of the hollow member or outersurface of the rotor during operation.

SUMMARY OF THE DISCLOSURE

According to one embodiment of the disclosure, an annular seal generallyincludes a hollow member having two ends and an inner surface. Thehollow member has a number of depressions formed in and extending aroundits inner surface. Each of the depressions having a depth that is afunction of its distance from one of the two ends.

Some embodiments of the present disclosure may provide numeroustechnical advantages. A particular technical advantage of one embodimentmay comprise an annular seal that exhibits enhanced damping over arelatively broad frequency range. The variable depth of the depressionsenables unique tailoring of this damping in order to alleviate theadverse effects of operation at various revolution speeds.

Although a specific advantage has been disclosed hereinabove, it will beunderstood that various embodiments may include all, some, or none ofthe previously disclosed advantages. Other technical advantages maybecome readily apparent to those skilled in the art of annular seals.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention and theadvantages thereof, reference is now made to the following briefdescriptions, taken in conjunction with the accompanying drawings anddetailed description, wherein like reference numerals represent likeparts, in which:

FIG. 1 is a partial cut-away view showing one embodiment of aturbomachine that may be implemented with annular seals according to theteachings of the present disclosure;

FIG. 2 is a perspective view of one embodiment of an annular seal thatmay be implemented on the turbomachine of FIG. 1;

FIG. 3 is a cut-away view showing several embodiments of annular sealsthat may each have a number of holes, the depth of which is a functionof its distance from one of its two ends;

FIG. 4A is a graph representing calculations that were performed toestimate the effective stiffness of the various annular seals of FIG. 3;

FIG. 4B is a graph representing calculations that were performed toestimate the effective damping on the various annular seals of FIG. 3;

FIGS. 5A-5F show several graphs including calculations and associatedmeasured test data for a particular variable hole depth annular seal ofFIG. 3 having a square root holed depth function that decreases alongits direction of flow;

FIGS. 6A-6F show several graphs including test data depictingnondimensionalized, measured rotordynamic coefficients for the variablehole depth (VHD) annular seal of FIG. 3 and a conventional constant holedepth (CHD) seal;

FIG. 7 is a graph comparing the relative leakage per pressure ratio ofthe variable hole depth (VHD) annular seal having a square root holeddepth function that decreases along its direction of flow with aconventional constant hole depth (CHD) annular seal; and

FIGS. 8A-8D are graphs showing comparisons between measurements andpredictions for the constant hole depth (CHD) annular seal forcomparable speed, preswirl, and clearance conditions showing comparableunder predictions for stiffness coefficients (k) and dampingcoefficients (C).

DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS

As previously described, the dynamic characteristics of annular sealscause dynamic resistance to the movement of the fluid such that asealing action occurs during operation. Gas compressibility in thedepressions of the surface of the annular seal function to reduce theeffective acoustic velocity of the adjacent gas. This action may producea strong frequency dependent dynamic reaction force characteristic inthe annular seal. Moreover, the seal's influence on the rotor'svibration can be modeled by stiffness and damping coefficients, andthese coefficients depend strongly on the excitation frequency. Theseals' stiffness and damping coefficients can have a pronounced effecton the compressor's stability of lateral motion.

FIG. 1 shows one embodiment of a turbomachine 10 that may be implementedwith annular seals according to the teachings of the present disclosure.In the illustrated example, turbomachine 10 generally includes one ormore impellers 12 that are disposed in a housing 14 and rotatinglycoupled to a rotor 16. Chambers 18 may be configured upstream anddownstream and the front and both sides of each of the impellers 12 tofacilitate controlled movement of fluids through the impellers 12. Toprevent fluid leakage from these chambers 18, annular seals 20 accordingto the teachings of the present disclosure may be incorporated.According to the teachings of the present disclosure, these annularseals may be provided with a number of depressions, each having a depththat varies along its axial extent for dynamic tailoring of variousdynamic characteristics of the turbomachine 10.

Certain embodiments of the annular seal 20 having a number ofdepressions that varies in depth along its axial extent may provide anadvantage in that the annular seal 20 may be tailored to improve itsrespective damping coefficient and/or stiffness through a relativelybroad frequency range. That is, selection of the depth of each of thedepressions along its axial extent may provide one approach to tailoringthe operating characteristics of the annular seal 20 for enhancedoperation on turbomachine 10.

The turbomachine 10 may be any rotating machine on which annular seals20 may be implemented. In the particular embodiment, turbomachine 10 isa centrifugal compressor; however, annular seals 20 according to theteachings of the present disclosure may also be implemented on otherturbomachines, such as axial-flow compressors or turbines. In thecentrifugal compressor shown, impellers 12 are arranged in a cascadingfashion such that one chamber 18 may couple the output of one impellerto the input of another.

The annular seals 20 may be any suitable type of annular seal 20implemented on turbomachine 10. The turbomachine 10 may have severalelements, such as a rotor 16, one or more impellers 12, and/or balancepistons 22 that may be collectively referred to as rotating members. Theannular seals 20 may be configured with any rotating member of theturbomachine 10 for providing a sealing action to moving fluids. Asshown in this particular embodiment, annular seal 20 a, generallyreferred to as a shaft seal, is disposed adjacent the rotor 16. Inanother embodiment, annular seal 20 b may be configured with a balancepiston 22 in which the annular seal 20 b may be referred to as a balancepiston seal when used with a straight-through compressor ordivision-wall seal when used with a back-to-back compressor. In anotherembodiment, annular seals 20 c may be configured on the front shroudportion of impellers 12 and may refer to eye packing annular seals.

FIG. 2 is an enlarged, disassembled view of one embodiment of annularseal 20 that may be used in conjunction with the turbomachine 10 ofFIG. 1. The annular seal 20 generally includes a hollow member 30 withan inner surface 32 having a number of depressions in the form of holesthat extend over the inner surface 32. The inner surface 32 may bedimensioned to receive a portion of the rotor 16 having a outer surface34. As will be described in detail below, annular seal 20 may beoperable to provide enhanced operating characteristics over other knownannular seal designs.

Hollow member 30 may form any portion of the stator of turbomachine 10.That is, hollow member 30 may form any portion of turbomachine 10 thatremains generally stationary relative to the rotor portion 16. Forexample, hollow member 30 may be integrally formed with the housing 14or may exist as a replaceable component of the housing 14.

Rotor portion 16 may be any portion of the rotor 16 of turbomachine 10such as described above. When configured within the hollow member 30,the rotor portion 16 is generally co-axially aligned with the innersurface such that the rotor portion 16 may freely rotate in hollowmember 30 without making physical contact. Annular seal 20 has an axialextent L that extends along the axis of the hollow member 30 and isbounded by two ends 36.

The inner surface 32 may have any suitable shape that allows the rotorportion 16 to rotate freely inside. In one embodiment, inner surface 32may have a generally cylindrical shape. In another embodiment, innersurface 32 may have a tapered shape such that its clearance from therotor portion 16 at one end 36 may be different from its clearance atits other end 36. Certain embodiments may provide of the presentdisclosure may provide an advantage in that the varying hole depth maybe significantly greater than the variation in hole depth provided bythe tapered shape of the inner surface 32 and therefore, may provideenhanced tailoring of the operating characteristics of the annular seal20.

In one embodiment, annular seal 20 is a hole pattern seal. A holepattern seal generally refer to a type of annular seal havingdepressions 38 in the form of generally round-shaped holes. These holepattern annular seals 20 may be configured to have a hole-area densityof up to approximately 79%. In another embodiment, annular seal 20 is ahoneycomb (HC) seal. A honeycomb seal generally refer to a type of sealin which the depressions 38 may be generally hexagonal-shaped holes thatare formed into a lattice-shaped structure. In this particularembodiment shown, a number of depressions 38 are formed in the innersurface 32 of hollow member 30.

According to the teachings of one embodiment of the present disclosure,each of the plurality of depressions 38 has a depth which is a functionof its distance from one of the ends 36. Annular seals 20 of this typemay be generally referred to as variable hole depth (VHD) annular seals.The depth of each of the depressions 38 relative to the otherdepressions 38 may provide another approach to tailoring the operatingcharacteristics of the annular seal 20 in certain embodiments. Certainembodiments of the present invention incorporating depressions 38 thatvary in depth along its axial extent may provide an advantage bytailoring the effective damping and effective stiffness coefficients ofthe annular seal 20 to reduce adverse effects of operation at varyingspeeds and loads.

Turbomachines 10 such as injection compressors require comparativelylong annular seals 20 with high pressure drops that have a significantimpact on rotordynamics. For example, balance-piston annular seals 20 bfor straight-through compressors may absorb the full head rise of theturbomachine 10. The division-wall annular seal 20 a incorporatingback-to-back impellers, may take about one half of the turbomachine'shead rise but deals with higher density gas. Hole pattern annular seals20 using depressions 38 in the form of round-shaped holes have beenrecently adopted for use in some oil and gas compressors. Known annularseals made of stainless steel have been replaced by hole pattern annularseals that are made of aluminum due to their ease of manufacture. Thus,annular seals 20 of the hole pattern type may provide enhanced operatingcharacteristics with respect to known annular seals in certainembodiments.

In one embodiment, honeycomb annular seals 20 are made from hightemperature stainless steel or aluminum. The honeycomb annular seal 20may be made from an electrical-discharge machining process or othersuitable approach.

Equation (1) is one embodiment of a two-control-volume model forcalculating the operating characteristics of the annular seal 20.

$\begin{matrix}{{- \begin{Bmatrix}{f_{sX}(s)} \\{f_{sY}(s)}\end{Bmatrix}} = {\begin{bmatrix}G & E \\{- E} & G\end{bmatrix}\begin{Bmatrix}{X(s)} \\{Y(s)}\end{Bmatrix}}} & (1)\end{matrix}$

Here, s is the Laplace domain variable, f_(s) the reaction force vector,and X(s), Y(s) are the Laplace-domain components of the relativedisplacement vector between the rotor portion 16 and hollow member 30.

In terms of frequency-dependent stiffness and damping coefficients, theseal model can be approximated by equation (2).

$\begin{matrix}{= {\begin{Bmatrix}{f_{sX}(s)} \\{f_{sY}(s)}\end{Bmatrix} = {{\begin{bmatrix}{K(\Omega)} & {k(\Omega)} \\{- {k(\Omega)}} & {K(\Omega)}\end{bmatrix}\begin{Bmatrix}X \\Y\end{Bmatrix}} + {\begin{bmatrix}{C(\Omega)} & {c(\Omega)} \\{- {c(\Omega)}} & {C(\Omega)}\end{bmatrix}\begin{Bmatrix}\overset{.}{X} \\\overset{.}{Y}\end{Bmatrix}}}}} & (2)\end{matrix}$

As can be seen, the two models are related by the terms,G(jΩ)=K(Ω)+jC(Ω) and E(jΩ)−k(Ω)+jc(Ω) where j=√{square root over (−1)}and Ω is the rotor precession frequency. In comparing the rotordynamicperformance of seals, the effective stiffness and damping coefficients,can be approximated by equations (3).

K _(eff)(Ω)=K(Ω)+c(Ω)Ω,C _(eff)(Ω)=C(Ω)−k(Ω)/Ω,   (3)

The effective damping coefficient (C_(eff)) combines the stabilizingdirect damping coefficient (C) and the destabilizing cross-coupledstiffness coefficient (k). These definitions apply only for small motionabout a centered position.

The two-control-volume model described above predicted that annularseals 20 had strong frequency dependent stiffness and dampingcoefficients, and tests results have generally confirmed thesepredictions. The effective damping coefficient (C_(eff)) of the annularseal combines the destabilizing influence of the cross-coupled stiffnesscoefficient (k) and the stabilizing influence of direct damping (C).Effective damping coefficient (C_(eff)) is negative at low rotorprecession frequencies due to stiffness coefficient (k) and positive athigher frequencies due to (C). The frequency at which it changes sign iscalled the cross-over frequency (Ω_(co)). In applications, cross-overfrequency (Ω_(co)) needs to be lower than the rotor's 16 first naturalfrequency (Ω_(nl)). Otherwise, the annular seal 20 may becomedestabilizing.

From a rotordynamics viewpoint, a decrease in cross-over frequency(Ω_(co)) and an increase effective damping coefficient (C_(eff)) may bebeneficial. The equations above show that gas compliance provided byhoneycomb or hole pattern annular seals 20 causes a reduction in thelocal effective acoustic velocity. Hence varying the hole depth axiallycould be expected to modify the two dimensional acoustic mode shapeswithin the annular seal and might favorably modify its rotordynamiccharacteristics. Thus, equations (1), and (2) show that operatingcharacteristics, such as cross-over frequency (Ω_(co)) and effectivedamping coefficient (C_(eff)) may be manipulated by varying the depth ofthe depressions 38 along the axial extent of the annular seal 20.

Calculations

Calculations were obtained using a model that was similar to thetwo-control-volume model of equations (1) and (2) that allowed for avariable hole depth function H_(d)(Z). The analysis was applied to anexisting hole pattern seal having a radius (R) of 57.37 millimeters(mm), a length (L) of 86.06 millimeters (mm), a clearance radius (Cr) of0.20 millimeters (mm), a supply pressure of 70 bars, and a pressureratio of 0.5.

FIG. 3 shows cross-sectional views of several embodiments of annularseals 20, each having particular hole-depth functions H_(d)(Z) that wereused in the calculations. Hole pattern annular seal 20 c has a linearhole-depth function that increases along the direction of flow 40. Holepattern annular seal 20 d has a linear hole-depth function thatdecreases along the direction of flow 40. Hole pattern annular seal 20 ehas a squared hole-depth function that increases along the direction offlow 40. Hole pattern annular seal 20 f has a squared hole-depthfunction that decreases along the direction of flow 40. Hole patternannular seal 20 g has a square root hole-depth function that increasesalong the direction of flow 40. Hole pattern annular seal 20 h has asquare root hole-depth function that decreases along the direction offlow 40. Hole pattern annular seal 20 i has a cosine hole-depth functionthat increases along the direction of flow 40. Hole pattern annular seal20 j has a cosine hole-depth function that decreases along the directionof flow 40. Hole pattern annular seal 20 k has a arc-cosine hole-depthfunction that increases along the direction of flow 40. Hole patternannular seal 20 l has a arc-cosine hole-depth function that decreasesalong the direction of flow 40.

Each of the annular seals 20 c through 20 l has holes, the depth ofwhich varies as a function of its distance from one of its two ends 36.In one embodiment, holes 38 that are equidistant from one of its twoends 36 may have a relatively similar depth. In other embodiments, holes38 that are equidistant from one of its two ends 36 may have a depthsthat vary from one another.

FIG. 4A shows calculated values that were obtained for the effectivestiffness coefficient (K_(eff)) for all variable hole depth (VHD)designs. Table 1 shows, in descending order for the effective stiffnesscoefficient (K_(eff)), several annular seals from FIG. 3 for anondimensionalized excitation frequency range of 0.5.

TABLE 1 Relative Hole Depth Depth Relative Performance Function ToDirection of (Descending Order) (H_(d)(Z)) Flow 1 QUADRATIC INCREASING 2SQUARE ROOT DECREASING 3 COSINE INCREASING 4 LINEAR INCREASING 5ARC-COSINE INCREASINE

FIG. 4B is a graph showing calculated values for the resultant effectivedamping coefficient (C_(eff)) for the variable hole depth annular seals20 of FIG. 3 having an inlet hole depth (H_(din)) of 3.30 millimeters(mm) zero preswirl, and running speed (ω) of 20,000 revolutions perminute (rpm). In descending order, the best performers for reducingcross-over frequency (Ω_(co)) are shown in Table 2.

TABLE 2 Relative Hole Depth Depth Relative Performance Function ToDirection of (Descending Order) (H_(d)(Z)) Flow 1 COSINE DECREASING 2SQUARE DECREASING 3 LINEAR DECREASING 4 ARC-COSINE DECREASING 5QUADRATIC DECREASING 6 QUADRATIC INCREASING 7 CONSTANT —

Generally, decreasing the hole depth axially along the direction of flow40 may be productive in reducing (Ω_(co)). All of the functions thatgive reduced cross-over frequency (Ω_(co)) also yield marked overallincreases in effective damping coefficient (C_(eff)). As shown, annularseal 20 h having a square root hole depth function H_(d)(Z) thatdecreases along its direction of flow 40 is predicted to give about 2.7times greater peak value for effective damping coefficient (C_(eff))than the constant hole depth design.

As can be seen from FIGS. 4A and 4B, the square root annular seal 20 hhaving depressions that decrease in depth along its direction of flowproduces both better damping characteristics and higher effectivestiffness coefficient (K_(eff)).

EXAMPLES

Based on the above calculations, the square root-decreasing hole-depthfunction annular seal 20 h was selected for testing. A test apparatuswas used to test the sample annular seal 20 h having a square root holedepth function that decreases in depth along its direction of flow 40.The test apparatus used external shakers to excite a softly-mountedhollow member 30 that contains two identical annular seals 20 h and issupported by two generally stiff hybrid bearings. Transient measurementscapture components of the input excitation force, the statoracceleration, and the relative displacement between the hollow member 30and rotor portion 16. Starting with these measurements, afrequency-domain parameter identification approach is employed toidentify the Hij complex dynamic stiffness coefficients using equation(5).

$\begin{matrix}{{- \begin{Bmatrix}F_{X} & {{- m_{s}}A_{X}} \\F_{Y} & {{- m_{s}}A_{Y}}\end{Bmatrix}} = {{- \begin{bmatrix}H_{XX} & H_{XX} \\H_{YX} & H_{YY}\end{bmatrix}}\begin{Bmatrix}D_{Y} \\D_{Y}\end{Bmatrix}}} & (5)\end{matrix}$

where F, A, D, are complex fast fourier transforms (FFTs) of the appliedforce, stator acceleration, and relative stator-rotor displacementvectors, respectively. Diagonal coefficients (Hij) are “direct”dynamic-stiffness coefficients. Off-diagonal terms are “cross-coupled”dynamic stiffness coefficients. The test apparatus uses alternate shakesin orthogonal directions to obtain direct and cross-coupled dynamicstiffness coefficients. For motion about the centered position, theorypredicts and measurements confirm that H_(XX)=H_(YY)=G andH_(XY)=−H_(yx)=E, for E and G as described and shown in equation (1).

Steady-state measurements in the test apparatus include rotor speed (ω)plus upstream and downstream temperatures and pressures. The availablesupply pressures reach 70 bars, and the seal pressure ratios was variedindependently of supply pressure. Tests were conducted at the followingthree pressure ratios: 0.3, 0.4, 0.5. The test variable hole depth (VHD)annular seal 20 h produced a pronounced negative static stiffness thatlimited test conditions to a supply pressure of no more than 55 bars. Itmay be noted that division-wall or balance-piston seals of compressorsnormally operate at pressure ratios near 0.5 and thus, tests conductedat a supply pressure of 55 bars provide a generally realistic assessmentof performance. The variable hole depth (VHD) annular seal 20 h wastested with three different preswirl inserts, producing a range ofpre-swirl circumferential velocity ratios. Since all current andproposed compressor applications employ swirl brakes, only zero-preswirldata was reviewed. Finally, tests were conducted at 10200, 15200, and20200 rotations per minute (rpm), but most of the results to bepresented were tested at 20200 rpm to get surface velocities that arenearer to actual compressor applications.

The tests were administered to measure rotordynamic coefficients of theannular seal 20 h. However, the test procedure measured stiffness anddamping arising from other elements, such as exit seals, hoseconnections, and the like. To account for these additional elements,base-line tests were conducted with the annular seal 20 h removed. Thesebase-line tests were conducted at reduced supply pressures to match theactual test back pressures experienced by the back-pressure seals. Oneset (H_(XX), H_(YY), H_(xy) and H_(yx)) of frequency-dependent dynamicstiffness coefficients were obtained as the average of thirty-twoseparate shake tests, which were averaged in the frequency domain.

To estimate the variability of dynamic data, ten consecutive tests wereconducted at 15,200 rpm. During these tests, the operating conditionsare held as constant as possible. The dynamic-stiffness data werereduced, and the standard deviation of the dynamic stiffness androtordynamic coefficients were obtained for discrete frequencies. Thestandard deviation obtained from these ten sequential tests defined theuncertainty at each frequency for both the baseline tests and testsconducted with the annular seals 20 h. Uncertainties in the dynamicstiffness coefficient results varied with frequency. For example,results near the line frequency of 60 Hertz (Hz) or multiples of thisfrequency are consistently poor. These frequencies were, therefore,avoided in creating the excitation waveform. Also, data at or nearrunning speed are poor and are discarded. The test uncertainty iscalculated at each frequency as the square root of the sum of thesquares of baseline uncertainty and seal test uncertainty at eachfrequency. The average uncertainty is displayed on the plots of FIGS.5A-5F as error bars for the rotordynamic coefficients. The uncertaintybars reflect+/−one standard deviation.

The first issue of interest concerns the accuracy of predictions for theannular seal 20 h with varying hole depth. FIGS. 5A-5F illustratemeasurements and predictions for the variable hole depth (VHD) annularseal 20 h having a square root hole depth function that decreases alongthe direction of flow at 20,200 rpm for zero preswirl, inlet pressure(P_(in)) of 41.4 bars, and pressure ratio (PR) of 0.5. The zero preswirlcondition is used in calculations and tests for the variable hole depthannular seal 20 h because of prior test results and field experiencesthat have shown balance-piston or division-wall seals need swirl brakes(or shunt injection) to minimize the cross-coupled stiffness coefficient(k). Starting with stiffness coefficient (K(Ω)), the predicted andmeasured functions are qualitatively similar. The static stiffness isnegative and fairly large versus predictions for small and positive. Therelative uncertainties were reasonable. In this case, effectivestiffness coefficient (K_(eff)(Ω)) basically coincides with stiffnesscoefficient (K(Ω)). For stiffness coefficient (k(Ω)), measurements andpredictions are in qualitative agreement, but measured values areconsistently higher by an approximate factor of 1.7 at lowerfrequencies. Relative uncertainties are reasonable.

Measured and predicted functions for damping coefficient (C(Ω)) arequalitatively similar, but measured damping values are higher by anapproximate factor of 2. For cross-coupled damping coefficient (c(Ω)),the measurements and predictions agree qualitatively, but the measuredvalues are higher. The relative uncertainties are higher forcross-coupled damping coefficient (c(Ω)), but this reflects the lowermagnitudes for this function, more than higher absolute uncertainty.Effective damping coefficient (C_(eff)(Ω)) is qualitatively wellpredicted. The cross-over frequency (Ω_(co)) is closely predicted. Above(Ω_(co)), measured effective damping is higher than predicted by anapproximate factor of 2.2.

The following nondimensionalized and normalized stiffness (K) anddamping coefficients (C) will be used to compare the present variablehole depth annular seal 20 h to a constant hole depth (CHD) seal.Equations (6) may be used to approximate stiffness, cross-coupledstiffness, and damping nondimensionalized comparisons between variablehole depth (VHD) annular seals 20 h and known constant hole depth (CHD)annular seals.

$\begin{matrix}{{K^{+} = {K( \frac{C_{F}}{{DL}\; \Delta \; P} )}},{k^{+} = {k( \frac{C_{F}}{{DL}\; \Delta \; P} )}},{C^{+} = {{{C( \frac{C_{F}}{{DL}\; \Delta \; P} )}\varphi} = {\frac{\overset{.}{m}}{{\,_{\pi}D}\; C_{Y}}\sqrt{\frac{R_{c}T_{m}}{2\Delta \; {PP}_{m}}}}}}} & (6)\end{matrix}$

With this approach, normalized damping coefficients have dimensions ofseconds. The nondimensionalization of equations (6) are generallyeffective for hole pattern (HP) and honeycomb (HC) annular seals.

FIGS. 6A-6F show several graphs providing nondimensionalized, measuredrotordynamic coefficients for the variable hole depth (VHD) annular seal20 h and a constant hole depth (CHD) annular seal. The two annular sealsare identical except the constant hole depth (CHD) seal had a hole depth(H_(d)) of 3.30 millimeters (mm). For both annular seals, running speed(ω) of 20,200 rpm, preswirl is zero, and the pressure ratio (PR) of 0.5.The variable hole depth (VHD) annular seal 20 h had an inlet pressure(P_(in)) of 55.2 bars, the constant hole depth (CHD) annular seal had aninlet pressure (P_(in)) of 70 bars. The stiffness coefficient (k*(Ω))and damping coefficient (c*(Ω)) basically coincide for the variable holedepth (VHD) annular seal 20 h and constant hole depth (CHD) annularseal. For stiffness coefficient (K*(Ω)), the variable hole depth (VHD)annular seal 20 h starts with a large negative value versus the constanthole depth (CHD) annular seal's relatively low positive value. Thevariable hole depth (VHD) annular seal's stiffness coefficient (K*(Ω))increases rapidly and catches up with the constant hole depth (CHD)annular seal at approximately 200 Hertz (Hz). The effective stiffnesscoefficient (K*_(eff)(Ω)) function closely resembles the stiffnesscoefficient (K*(Ω)) function for both the variable hole depth (VHD)annular seal 20 h and the constant hole depth (CHD) annular seal.

The variable hole depth (VHD) annular seal 20 h has a relatively higherlow-frequency value for damping coefficient (C*(Ω)) than the constanthole depth (CHD) seal. The variable hole depth (VHD) annular seal 20 hand the constant hole depth (CHD) annular seals have about the samevalues at 300 Hertz (Hz). For effective damping coefficient(C_(eff)*(Ω)), the cross-over frequency (Ω_(co)) values for the constanthole depth (CHD) annular seal and variable hole depth (VHD) annular seal20 h are, respectively, 50 and 70 Hz, reflecting a reduction of about40%. The peak values for effective damping coefficient (C_(eff)*(Ω)) forthe constant hole depth (CHD) and variable hole depth (VHD) annularseals are, respectively, 7.42 E-05 seconds at 160 Hertz (Hz) and 10.8E-05 seconds at 100 Hertz (Hz), reflecting an increase in peak effectivedamping by a factor of about 1.60.

FIG. 7 provides a comparison of the constant hole depth (CHD) andvariable hole depth (VHD) annular seals leakage coefficient, versuspressure ratio. The leakage coefficient (O) may be calculated accordingto equation (7).

$\begin{matrix}{\varphi = \frac{\overset{.}{m}}{\pi \; D\; {C_{r}( \sqrt{\frac{R_{c}T_{m}}{2\Delta \; {PPm}}} )}}} & (7)\end{matrix}$

In the leakage-coefficient definition, m dot is the mass flow rate,R_(c) is the gas constant, and T_(in) is the inlet temperature. As canbe seen, the variable hole depth (VHD) annular seal 20 h leaks less thanthe constant hole depth (CHD) annular seal. The difference decreaseswith increasing pressure ratio (PR).

The test results show that the variable hole depth (VHD) annular seal 20h can produce improved damping characteristics in terms of a reducedcross-over frequency and increased effective damping. The measuredincrease in peak effective damping for the variable hole depth (VHD)annular seal 20 h was around 1.6 versus predictions of 2.7

One particular observation to be made regarding the perceived increasein effective damping of the variable hole depth (VHD) annular seal 20 his that the model under predicts the dynamic stiffness coefficients forboth seals by about the same amount for the test conditions shown. FIGS.8A-8D provide comparisons between measurements and predictions for theconstant hole depth (CHD) annular seal for comparable speed, preswirl,and clearance conditions, showing comparable under predictions forstiffness coefficients (k) and damping coefficients (C). Thus, the modeldoes a relatively better job with non-zero preswirl conditions.

The pronounced negative static stiffness shown by the variable holedepth (VHD) annular seal 20 h was not predicted and limited the feasibletest supply pressures, (ΔPs), and clearances. Beyond a limiting supplypressure (ΔP), the test rig stator would “stick” to the test rotor andcould not be pulled away using the hydraulic shakers. Each shaker hasapproximately 4400 newton (N) static capability. This phenomenon andrestriction have appeared previously while testing one honeycomb (HC)seal and several hole pattern (HP) annular seals. The constant holedepth (CHD) annular seal, with hole depth (H_(d)) of 3.30 millimeters(mm) was tested at full pressure at the two radial clearances of 0.10and 0.20 millimeters (mm). An identical annular seal, except for asmaller hole depth (H_(d)) of 3.18 millimeters (mm), could not be testedat full pressure at the 0.10 millimeters (mm) radial clearance becauseof negative stiffness. Because of the negative stiffness, the presentvariable hole depth (VHD) annular seal 20 h could not be testedeffectively at any reasonable supply pressure for the reduced 0.10millimeter (mm) radial clearance.

Seal divergence is known to cause negative stiffness. The dimensions ofthe variable hole depth (VHD) annular seal 20 h was measured after beingfitted into the test housing, and no measurable divergence could bedetected. During tests, the axial pressure distribution tends to createlarger clearances at the inlet leading to a converging flow path ratherthan a diverging flow path. Also, the gas expansion creates asignificant drop in temperature as the gas moves down the seal. Again,this temperature distribution would cause a convergent flow conditionwith tighter exit clearances. The “friction-factor jump” phenomenonprovides the most reasonable explanation for negative stiffness.Specifically, flat-plate test results for which the friction factorincreases with increasing Reynolds numbers at elevated Reynolds numbersin the range of 20000.

The “Lomakin” effect predicts a positive stiffness of annular seals,resulting from the interaction of the inlet loss and friction factorwhen the friction-factor decreases with increasing Reynolds number,∂ff/∂R_(e)>0, and a reduced or negative stiffness for the oppositebehavior. The fact that the present variable hole depth (VHD) annularseal 20 h leaks less than the constant hole depth (CHD) annular sealsupports the idea that the variable hole depth (VHD) annular seal 20 hhas friction-factor jump behavior. Since the model does not account forfriction-factor jump phenomenon, its failure to predict the pronouncednegative direct stiffness of FIGS. 5A-5F is understandable. Note that nomethod exists today for predicting the occurrence of friction-factorjumps due to cavity flow phenomena for constant hole depth (CHD) orvariable holed depth (VHD) geometries.

The negative static stiffness produced by the variable holed depth (VHD)annular seal 20 h was a problem for this test program, because thestator was softly mounted. However, predictions suggest that negativestiffness may not be a problem in compressors because of the much higherbearing-rotor stiffness. Moreover a small convergent taper can probablyproduce sufficient additional stiffness to offset any concerns due tothe negative stiffness of the annular seal 20 h.

The results produced here with a variable hole depth (VHD) annular seals20 h are not predicted for deeper or shallower constant hole depth (CHD)annular seals. The variable hole depth (VHD) configurations couldreadily be realized with honeycomb hole patterns that are made viaelectrical discharge machining, rather than standard stainless steelhoneycomb (HC) surfaces.

Although the present disclosure has been described in severalembodiments, a myriad of changes, variations, alterations,transformations, and modifications may be suggested to one skilled inthe art, and it is intended that the present disclosure encompass suchchanges, variations, alterations, transformations, and modifications asfalling within the spirit and scope of the appended claims.

I claim:
 1. A seal, comprising: a hollow member having a longitudinalaxis extending between a first axial end and a second axial end thereof;and a plurality of depressions formed in an inner surface of the hollowmember, each depression of the plurality of depressions having arespective depth measured from the longitudinal axis to a respectivebottom of each depression of the plurality of depressions, therespective depth of each depression of the plurality of depressionsprogressively varying from the first axial end toward the second axialend as a differentiable function of an axial distance from the firstaxial end.
 2. The annular seal of claim 1, wherein the hollow member iscylindrical.
 3. The annular seal of claim 1, wherein the inner surfaceis tapered.
 4. The annular seal of claim 1, wherein the inner surfacemaintains a nominally constant distance from the longitudinal axis ofthe hollow member.
 5. The annular seal of claim 1, wherein the firstaxial end and the second axial end define a length of the hollow member,and the plurality of depressions are disposed along substantially thelength of the hollow member.
 6. The annular seal of claim 1, wherein thedifferentiable function is a differentiable linear function, such thatthe respective depth of each depression of the plurality of depressionsincreases linearly from the first axial end to the second axial end. 7.The annular seal of claim 1, wherein the differentiable function is adifferentiable exponential function, such that the respective depth ofeach depression of the plurality of depressions increases exponentiallyfrom the first axial end to the second axial end.
 8. The annular seal ofclaim 1, wherein the differentiable function is a differentiablesinusoidal function, such that the respective depth of each depressionof the plurality of depressions increases sinusoidally from the firstaxial end toward the second axial end.
 9. The annular seal of claim 1,wherein the differentiable function is a differentiable sinusoidalfunction, such that the respective depth of each depression of theplurality of depressions decreases sinusoidally from the first axial endtoward the second axial end.
 10. The annular seal of claim 1, whereinthe differentiable function is a differentiable square root function.11. The annular seal of claim 1, wherein at least some of the pluralityof depressions are round-shaped.
 12. The annular seal of claim 1,wherein at least some of the plurality of depressions arepolygonal-shaped.
 13. The annular seal of claim 1, wherein the innersurface has a depression-area density of approximately 79%.
 14. Aturbomachine, comprising: the annular seal of claim 1; and a rotorco-axially aligned with the hollow member along the longitudinal axisand configured to rotate within the hollow member, the annular sealconfigured to increase a damping coefficient of the turbomachine.
 15. Aseal for a turbomachine, comprising: an annular member having alongitudinal axis extending between a first axial end and a second axialend thereof, the first axial end and the second axial end defining anaxial length of the annular member; and a plurality of depressionsextending radially outward from an inner surface of the annular member,the plurality of depressions disposed along substantially the axiallength of the annular member, each depression of the plurality ofdepressions having a respective depth measured from the longitudinalaxis, the respective depth of each depression of the plurality ofdepressions progressively varying from the first axial end to the secondaxial end as a differentiable function of an axial distance from thefirst axial end, the differentiable function of the axial distance is adifferentiable exponential function, a differentiable sinusoidalfunction, or a differentiable square root function.
 15. The seal ofclaim 14, wherein the first axial end is an inlet end and the secondaxial end is an outlet end.
 16. The seal of claim 14, wherein the firstaxial end is an outlet end and the second axial end is an inlet end. 17.The seal of claim 14, wherein the inner surface is disposed at anominally constant distance from the longitudinal axis of the annularmember.
 18. A turbomachine, comprising: a rotor supported in a housingof the turbomachine; and a seal disposed about an outer surface of therotor and configured to increase a damping coefficient of theturbomachine, the seal comprising: an annular member having alongitudinal axis extending between a first axial end and a second axialend thereof; and a plurality of depressions formed in an inner surfaceof the annular member, each depression of the plurality of depressionshaving a respective depth measured from the longitudinal axis, therespective depth of each depression of the plurality of depressionsprogressively varying from the first axial end to the second axial endas a differentiable function of an axial distance from the first axialend.
 19. The turbomachine of claim 18, wherein the inner surface of theannular member and the outer surface of the rotor define a clearancetherebetween, the clearance increasing from the first axial end to thesecond axial end of the annular member.
 20. The turbomachine of claim18, wherein the inner surface of the annular member maintains anominally constant distance from the outer surface of the rotor.